Extremal H-free planar graphs

Given a graph H, a graph is H-free if it does not contain H as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213 230], that is, how many edges can an H-free planar graph on n vertices have? We define ex(p) (n, H) to be the maximum number of edges in an H-free planar graph on n vertices. We first obtain several sufficient conditions on H which yield ex(p) (n, H) = 3n - 6 for all n >= vertical bar V(H)vertical bar. We discover that the chromatic number of H does not play a role, as in the celebrated Erdos-Stone Theorem. We then completely determine ex(p) (n, H) when H is a wheel or a star. Finally, we examine the case when H is a (t, r)-fan, that is, H is isomorphic to K-1 + tK(r-1), where t >= 2 and r >= 3 are integers. However, determining ex(p)(n, H), when H is a planar subcubic graph, remains wide open.